The Bailey Challenge

BY PROFESSOR EMERITUS

HERB BAILEY

We had nearly 125 solvers meeting the challenge of our fall issue problems. There were some familiar names and several new ones on the list. Hopefully, you will enjoy this issue’s challenges as well.

SPRING PROBLEM 1

The total cost of one chocolate bar and two identical packs of gum is $4.15. One chocolate bar cost $1 more than one pack of gum. Determine the cost of one chocolate bar. SPRING PROBLEM 2

A census taker arrived at the door of a grizzled Montanan and asked about the number of children the woman had and their ages. The woman had little use for census takers and the government in general and said simply, “I have three sons and the product of their ages is 72. The sum of their ages is my house number.“ She then stood silent. The census taker looked up at the house number and said, “That is not enough information.“ At this time, the woman stated, “The oldest can read.“ She then slammed the door. The census taker went away happy. How old were the children?

SPRING BONUS

If a circle is inscribed in a triangle with vertex angles 30, 60 and 90. What is the area of the triangle divided by the area of the circle? Hint: The center of the inscribed circle (incenter) is found at the intersection of the three angle bisectors. [Extra credit is awarded for solving without using trigonometry.]

FALL PROBLEM 1 SOLUTION

Michelle average (5, 10, 15, 16, 24, 28, 33, 37)=168/8=21 Daphne average=20=140/7. 168–140=28 removed

FALL PROBLEM 2 SOLUTION

There is no other solution than reflection/rotations. Since the center squares each touch six other squares, the center squares have to be the end numbers of the sequence: 1 and 8. That forces the upper and lower squares to be 2 and 7. Of the four outer squares, 3 and 4 have to be opposite (away from) 2 while, 5 and 6 have to be away from 7.

3

5 7

1 4

8 6

2

FALL BONUS PROBLEM 1 SOLUTION

There is a repeating pattern of remainders (R) for the series of 3n when divided by 5 (1,3,4,2). The pattern recurs in intervals of 4. So, 3n =30, 34, 38, 312, 316, and 320 all have remainder =1. Since these n values are all divisible by 4 and 10,000 is also evenly divisible by 4, the remainder when 310,000 is divided by 5 must also be 1.

Send your solutions to Herb.Bailey@rose-hulman.edu or to: Herb Bailey, 8571 Robin Run Way, Avon, IN 46123. Alumni should include their class year. Congratulations to the following solvers of the fall problems: ALUMNI: T. Jones, 1949; D. Camp, 1955; J. Moser, 1956; A. Sutton, 1956; P. Cella, 1958; D. Bailey, 1959; J. Kirk, 1960; W. Perkins, 1960; R. Archer, 1961; L. Hartley, 1961; J Ray, 1961; J. Tindall, 1961; R. Lovell, 1963; S. James, 1965; R. Kevorkian, 1966; R. Lowe, 1969; W. Myers, 1969; T. Winenger, 1969; J. Walter, 1969; J. Hightower, 1970; S. Jordan, 1970; J. Moehlmann, 1970; K. Moran, 1970; E. Arnold, 1971; D. Jordan, 1971; R. LaCrosse, 1971; W. Pelz, 1971; J. Witten, 1971; D. Hagar, 1972; G. Houghton, 1972; D. Moss, 1972; J. Sanders, 1972; R. Engelman, 1973; R. Kominiarek, 1973; M. Marinko, 1973; D. Van Laningham, 1973; J. Zumar, 1973; T. Rathz, 1974; D. Wheaton, 1974; D. Willman, 1974; W. Crellin, 1975; P. Eck, 1975; D. Copeland, 1976; B. Hunt, 1976; J. Schroeder, 1976; D. Walker, 1976; G. Matthews, 1977; T. Greer, 1978; R. Strickland, 1978; M. Clouser, 1979; R. Priem, 1979; J. Slupesky, 1979; S. Bagwell, 1980; M. Dolan, 1980; R. Joyner, 1980; J. Koechling, 1980; L. Smith, 1980; J. Farrell, 1981; S. Nolan, 1981; R. Roll, 1981; M. Taylor, 1982; R. Downs, 1983; S. Hall, 1983; J. Marum, 1983; B. Wade, 1983; G. Schafer, 1985; M. Solanki, 1985; C. Wilcox, 1985; G. Harding, 1986; C. Hastings, 1986; J. Vierow, 1986; M. Walden, 1986; D. Johnson, 1987; M. Lancaster, 1987; B. Seidl, 1987; S. Sarma, 1988; T. Sorauf, 1989; G. Heimann, 1990; P. Acevedo, 1991; R. Burger, 1991; J. Harris, 1991; R. Hochstetler, 1991; P. Kimmerle, 1991; R. Wilkinson, 1991; J. Zamora, 1991; R. Virostko, 1992; R. Antonini, 1993; W. Haas, 1993; T. Westbrook, 1996; W. Lewis, 1997; C. Mills, 1998; M. Pilcher, 1998; C. Ehrhart, 1999; B. Creel, 2000; T. Kibbey, 2003; G. Rahman, 2003; W. Casey, 2004; J. Somann, 2004; S. Tourville, 2005; R. Gulden, 2006; K. McCarthy, 2006; T. Homan, 2007; J. Krall, 2007; D. Schoumacher, 2010; M. Schoumacher, 2010; C. Drake, 2013; J. Althouse, 2015; J. Khusro, 2016; T. Mulc, 2016; Z. Watson, 2019; and J. Martin, 2020. FRIENDS: T. Cutaia, A. Foulkes, B. Harding, M. Holmes, P. Hines, J. Ley, M. Loper, J. Marks, J. Mathis, M. Moore, P. Nilsen, L. Puetz, E. Robertson, J. Robertson, C. Rozmaryn, R. Schoumacher, D. Shafer, L. Stafford, J. Walsh, E. Wern, and J. Wilcox.

WILKINSON MIXES MATH FORMULAS

As a research scientist, Royce Wilkinson enjoys the interface between chemistry and biology. He also takes delight in solving the mathematical Bailey Challenge problems in each Echoes issue and putting his answers on paper. For the past five years, the 1991 chemistry alumnus has submitted detailed solutions to show his enjoyment in the quarterly problems and support for the Challenge, a unique fixture of Rose-Hulman’s alumni magazine. “I usually work on the problems for a few minutes in the mornings as I am getting ready for work,“ says Wilkinson, an assistant research professor of microbiology and immunology at Montana State University. “Often, I will think about possible solutions over the next few days and try them out in the margins of Echoes. If I think I have the basic solutions figured out, I will spend some time on a Saturday morning writing up the solutions and then submitting them.“ Wilkinson enjoys tackling geometryoriented Challenge problems as well as those needing logic-based solutions. “As a biochemist, my daily math usually doesn't extend much beyond C1V1 = C2V2. So, it is nice to try solving things that push beyond some basic math skills,“ he says. After graduating from RHIT, Wilkinson earned a doctorate in a natural products lab isolating and characterizing bio-active secondary metabolites from fungi. Since then, he has worked in a wide variety of labs ranging from biochemistry to organic synthesis to immunology and virology. He currently is a researcher in MSU’s Wiedenheft Laboratory, studying bacterial CRISPR defense systems. “I enjoy the day-to-day bench work in a lab and the problem-solving that is a constant part of science when things don't work the way you expect or hope that they will work,“ he says.